Random simplices : from beta-type distributions to high-dimensional volumes / Zakhar Kabluchko, David Albert Steigenberger, Christoph Thäle.

Format:

Book

Author/Creator:

Kabluchko, Zakhar, 1982- author.

Steigenberger, David Albert, 1995- author.

Thäle, Christoph, 1984- author.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 2383.

Lecture notes in mathematics, 0075-8434 ; volume 2383

Language:

English

Subjects (All):

Simplexes (Mathematics).

Polytopes.

Stochastic geometry.

Convex geometry.

Physical Description:

xvi, 297 pages : illustrations (some color) ; 24 cm.

Place of Publication:

Cham, Switzerland : Springer, [2026]

Summary:

This book provides an introduction to the theory of random beta-type simplices and polytopes, exploring their connections to key research areas in stochastic and convex geometry. The random points defining the beta-type simplices, a class of random simplices introduced by Ruben and Miles, follow beta, beta-prime, or Gaussian distributions in the Euclidean space, and need not be identically distributed. A key tool in the analysis of these simplices, the so-called canonical decomposition, is presented here in a generalized form and is employed to derive explicit formulas for the moments of the volumes of beta-type simplices and to prove distributional representations for these volumes. Three independent approaches are described, including the original Ruben-Miles method. In addition, a version of the canonical decomposition for beta-type polytopes is provided, characterizing their typical faces as volume-weighted beta-type simplices. This is then applied to compute various expected functionals of beta-type polytopes, such as their volume, surface area and number of facets. The formulas for the moments of the volumes are also used to investigate several high-dimensional phenomena. Among these, a central limit theorem is established for the logarithmic volume of beta-type simplices in the high-dimensional limit. The canonical decomposition further motivates the study of beta-type distributions on affine Grassmannians, a subject to which the last chapter is dedicated. Largely self-contained, requiring minimal prior knowledge, the book connects these topics to a broad range of past and current research, serving as an excellent resource for graduate students and researchers seeking to engage with the field of stochastic and integral geometry.

Contents:

Chapter 1. Prologue

Part I. Introduction

Chapter 2. Beta-type distributions: Key properties and first applications

Chapter 3. Blaschke-Petkantschin formulas

Part II. Beta-type simplices

Chapter 4. Beta-type simplices and canonical decomposition of Ruben and Miles

Chapter 5. Volumes of beta-type parallelotopes

Chapter 6. Volumes of beta-type simplices

Chapter 7. Alternative approaches to volumes of beta-type simplices

Part III. Applications and further results

Chapter 8. Facets and volumes of beta-type polytopes

Chapter 9. Limit theorems for volumes of beta-type simplices

Chapter 10. Properties of beta-type distributions on affine Grassmannians.

Notes:

Includes bibliographical references (pages 283-293) and index.

ISBN:

9783032028631

3032028639

OCLC:

1586925282